Abstract

We consider a compact, oriented, smooth Riemannian
manifold $M$ (with or without boundary) and we suppose $G$ is a
torus acting by isometries on $M$. Given $X$ in the Lie algebra and
corresponding vector field $X_M$ on $M$, one defines Witten's
inhomogeneous coboundary operator $\d_{X_M} = \d+\iota_{X_M}:
\Omega_G^\pm \to\Omega_G^\mp$ (even/odd invariant forms on $M$) and
its adjoint $\delta_{X_M}$. Witten \cite{Witten} showed that the
resulting cohomology classes have $X_M$-harmonic representatives
(forms in the null space of $\Delta_{X_M} =
(\d_{X_M}+\delta_{X_M})^2$), and the cohomology groups are
isomorphic to the ordinary de Rham cohomology groups of the set
$N(X_M)$ of zeros of $X_M$. Our principal purpose is to extend these
results to manifolds with boundary. In particular, we define
relative (to the boundary) and absolute versions of the
$X_M$-cohomology and show the classes have representative
$X_M$-harmonic fields with appropriate boundary conditions. To do
this we present the relevant version of the Hodge-Morrey-Friedrichs
decomposition theorem for invariant forms in terms of the operators
$\d_{X_M}$ and $\delta_{X_M}$. We also elucidate the connection
between the $X_M$-cohomology groups and the relative and absolute
equivariant cohomology, following work of Atiyah and Bott. This
connection is then exploited to show that every harmonic field with
appropriate boundary conditions on $N(X_M)$ has a unique
$X_M$-harmonic field on $M$, with corresponding boundary conditions.
Finally, we define the $X_M$-Poincar\'{e} duality angles
between the interior subspaces of $X_M$-harmonic fields on $M$ with
appropriate boundary conditions, following recent work of DeTurck and Gluck.